Simple Harmonic Oscillation

Let us start with the simplest of observations: springs bounce. To analyze this phenomenon further, we will simplify the spring in the same spirit as the point particle approximation, and limit its motion to one dimension. Our one dimensional spring can be compressed or extended along the x axis, but cannot be bent or otherwise move in any direction perpendicular to its length.

The first thing that we notice about our spring is that it has a fixed length only when it is not being compressed or extended. This length can be called the spring's equilibrium length, l. It will be convenient to place the spring on the x axis so that one end is anchored at -l, and the other end lies exactly at x = 0 when the spring is at equilibrium.

Small Oscillations

The spring is a nearly universal model in physics. This is true in part because only it and the 1/r2 force can describe exactly periodic phenomena. As a result, we have the mathematical tools necessary to completely analyze such systems. But the primary utility of the spring goes back to our original observation: it bounces.

In words, this means that the rate of change of the rate of change of position is proportional to the position. Graphically, we can try to picture the position as an oscillatory function of time: perhaps a sine function:

At each point on that graph, the slope gives us the velocity of the mass at that time. The velocity graph must also be an oscillatory function of time:

and the slope at any point on this graph gives us the acceleration of the mass. Clearly, the graph of acceleration versus time must also be oscillatory, and to satisfy our equation, every point on it must be proportional to the value of the position at that time, but reflected about the x axis because of the minus sign:

The kinetic energy is always a maximum at the equilibrium position, where the potential energy is zero, and the potential energy is always a maximum at the extremes of the oscillation, where the velocity (and kinetic energy) is zero. Conservation of energy then tells us that the total energy of the oscillator is just the potential energy at the maximum displacement from equilibrium. This displacement is called the amplitude A, and the total energy is

In an
electrical circuit, charge
plays the same role as position does in mechanical systems; therefore current is analogous to velocity.
Since energy is voltage times charge, voltage must play the part of
mechanical force (since force is the gradient of energy). In fact,
electrical potential is often called electromotive force. We therefore have the following correspondences:

F = - k x

V = Q / C

U = k x2 / 2

U = Q2 / 2 C

Fdrag = - b V

V = I R

The first two correspondences would lead us to believe that the spring constant behaves like the inverse of the
capacitance.
However, consider a uniform mass suspended from the ceiling with two identical springs, one attached to each end of the mass.
In the mechanical system, both springs stretch the same displacement, and so we have

F = - k1 x - k2 x.

This means that two springs in parallel have an effective
spring constant which is simply the sum of the two spring constants.
If we connect the two springs in series (one on the end of the other)
and suspend our mass from the end, we have

F = - k1 x1

= - k2 x2

= - keffective (x1 + x2)

= - keffective (- F / k1 - F / k2)

which means that the spring constants of springs in series behave like capacitances:

keffective = 1 / (1 / k1 + 1 / k2)

The lesson here is that isomorphisms can be useful tools,
but the relationships must reflect the physical phenomenology:

charge / displacement

voltage / force

capacitors in series

same

split between them

springs in series

split between them

same

capacitors in parallel

split between them

same

springs in parallel

same

split between them

So the spring constant does behave like the inverse of the capacitance, but because the roles of charge and displacement,
voltage and force, behave oppositely in springs and capacitors, the equations for the effective spring constant of
springs in series and parallel are isomorphic to those for capacitance.

The next section introduces us to waves, and discusses the effects of boundary conditions and superposition on standing waves.